Quantum Energy of the Atom, expressed by the space
quantum, the time quantum
and the mass of the Universe

By Louis Nielsen, Senior Physics Master

Introduction

In the following I shall show that a connection exists between the
energy states of a hydrogen atom and the total content of energy of the
Universe. It will be shown that the mechanical energy of an electron in
a certain state is equal to a definite fraction of the total energy of
the Universe. It will be shown that the discontinuous properties are a
consequence of the quantization of space, time and mass.

In 1913 Niels Bohr showed (Niels Bohr (1885-1962): 'On the Constitution
of Atoms and Molecules', Philosophical Magazine, vol. 26, p. 1 (1913)),
how it was possible to quantize the energy of a H-atom, and thus give a
quantitative "explanation" of the experimental data concerning
the line spectra of the H-atom. To quantize the energy possibilities in
an H-atom, Bohr assumed ad hoc that the angular momentum of the electron,
viz. the product of its mass, velocity and distance from the proton, was
equal to a natural number multiplied by Planck's Constant divided
by 2*. By means of this ad hoc
supposition, Bohr could deduct a quantized energy formula, i.e. the
electron can only have quite definite and separate values of its energy.

However, the physical quantity we call the angular momentum is
calculated by the fundamental physical quantities, distance, time and
mass, which is also the case for all other physical quantities! As a
consequence it is reasonable to assume that the quantization is already
present at this fundamental level! I shall show that the quantization
at a "higher" level is a consequence of the quantization of space,
time and mass.Fundamentally we have:
The space quantum  elementary length r0 is given by:

(1)

where h is Planck's Constant, c0 is the velocity of light,
and M0 is the total matter/energy mass of the Universe.
The time quantum  elementary time t0 is given by:

(2)

The mass of an "elementary" particle is assumed to be a fraction
of the total mass of the Universe. Every finite physical distance
is a natural number  the
space quantum number multiplied by
elementary length.
It means that the following is valid:

(3)

Likewise every physical time interval is equal to a natural number  the time quantum number
multiplied by elementary time
t0. Thus the following is valid:

(4)

All physical quantities, such as velocity, acceleration, force, work,
energy etc., are defined as usual, but we must now take into consideration
the quantization of space, time and mass. This fundamental 'atomization'
implies that all movement is discontinuous, all movements take place in
'jumps'. Furthermore all physical processes in a definite system will
characteristically consist of discontinuous changes of certain physical
quantities.

Processes in a physical system can be described by the change of
certain quantum numbers, belonging to definite physical quantities. All
physical quantities can be expressed by the cosmically fundamental
quantities: the space quantum, the time quantum and the total mass of
the Universe.

Velocity and acceleration in the discontinuous movement

We define the velocity v of a particle in a definite reference
system in the usual way:

(5)

where is the distance travelled
in the time interval . In this
definition equation is the space
quantum number belonging to the distance , and the time quantum
number belonging to the time interval . The ratio between these two quantum numbers defines a
rational velocity quantum number, denoted by
. This velocity quantum number is
from the interval between 0 and 1, as no velocity surpasses c0.
If a particle changes its velocity in a certain time interval, the
particle is said to accelerate. The magnitude of this acceleration
is defined by the following expression:

(6)

In this definition equation, we shall call the acceleration quantum number which is seen to be a
rational number, lying in the interval 0 to 1. The ratio between the
velocity of light and elementary time c0/t0 gives
a theoretical upper limit for the acceleration of a particle. This upper
limit is denoted by amax.
If  in a certain reference system  it is registered that a
particle accelerates, we say that it is influenced by a force.

The quantum energy of the hydrogen atom.

In the following I shall show that the mechanical energy, viz. the sum
of the kinetic energy and the potential energy, of an electron, being in
a certain state of movement in the hydrogen atom, is quantized, and
that this quantum energy can furthermore be expressed by a certain
fraction of the total energy of the Universe.

We shall make the calculations on the following idealized model of
the hydrogen atom: we shall presume that the electron moves in a closed
'quantum circle' around a proton, which we shall presume is at rest in
a chosen reference system. We shall presume that the H-atom does not
interact with other partial systems, which of course is impossible in
the real Universe. In a first approximation we shall disregard magnetic
forces, caused by the movement of the electron in its orbit and own
rotation (spin) of the electron and the proton.

Furthermore we shall assume that both the electron and the proton are
mathematical points, which of course is not realistic. However, this
mathematical idealization has proved fruitful in other contexsts, f.i.
in the ideal gas model, in which it is also assumed that the particles
have no extension.

To make the model a little more physical, we can presume that both
the proton and the electron have an extension equal to elementary length.
Coulomb's electrostatic force law, which we shall use, is valid for
charges of mathematical points.

The quantum energy of the electron

The mechanical energy of the electron Em can be written:

(7)

where me is the mass of the electron at rest, v its discontinuous
velocity in a certain 'quantum orbit', kc is the Coulomb
Constant, e the electric elementary quantum, and r the average 'quantum
radius'.

In order that the electron shall be able to move in a closed orbit, it must
be influenced by a 'centripetal force', towards the proton. Thus we can
write Newton's 2nd Law for the movement of the electron as:

(8)

The right side of equation (8) is the expression for the electric Coulomb
force. By means of equation (8) we can transform equation (7) as:

(9)

Inserting the quantized velocity expression from equation (5) we get:

(10)

where the mass me of the electron is expressed as a whole
fraction of M0, the total mass of the Universe, thus:

(11)

where we shall call nm the mass quantum number.
We finally transform to:

(12)

where we can call ne the electron's cosmic energy quantum
number
From equation (12) we see the following: the quantum energy of an electron
in a certain quantum orbit is equal to a whole fraction of the total energy
of the Universe, which is logically reasonable. That the energy is negative
shows that the electron is in a 'bound' quantum orbit, and besides, the sign
is decided by the choice of origo for the electric potential energy.

As the light constant c0 is equal to the ratio between elementary
length and elementary time, we see the following from equation (12):
The quantum energy of the atom is determined by elementary length,
elementary time and the total energy of the Universe. To this comes a
quantum number, dependent of the used system. In equation (12) ne
is the cosmic energy quantum number of the atom.
Niels Bohr's quantum energy formula contains the mass of the electron,
its electric charge, Planck's Constant and the Coulomb Constant,
quantities characterized by the atom physical level, and not so fundamental
in their nature as the quantities I have used in my deduction.

As the total mass in the Universe is around 1.6 ·
1060 kg (see my quantum cosmology),
which is an extremely high figure, the cosmic energy quantum number is
also extremely high, in the order of 1095. Practical calculations
with such high figures are difficult, and we can therefore for practical
reasons again introduce the rest mass of an electron in equation (12).
My purpose of the preceding deductions has been to show the holistic
connections in our Universe. I shall formulate it as follows:
Everything determines everything!

Quantum Physical deduction of the Rydberg Formula.

In my treatise: Rational Quantum Physics, I have made a quantum physical
treatment of the empirically found
Rydberg Formula.
In this section I shall give a quantum physical deduction of the formula.
We consider two energy states of the atom, E1 and E2,
given by:

(13)

The ratio between these two energies is:

(14)

where ne,1/ne,2 gives the ratio between the
electron's cosmic energy quantum numbers in the two states. This is
furthermore given by:

(15)

i.e. the square of the ratio between the two rational velocity quantum
numbers, corresponding to the electron's two velocities.
We can now express the ratio between these two velocity quantum numbers
by space quantum numbers, which can only take natural numbers, viz. 1, 2,
3, etc.
If an electron is in a higher state of energy, E2, it will,
in a quantum jump 'seek' down to the lower energy E1. During
this process, conservation of angular momentum shall be valid, i.e. the
product of the mass of the electron, its velocity and its distance from
the center of force shall be constant. Assuming a photon is emitted
'radially', the following must be valid:

(16)

Expressing velocities and distances by quantized quantities, we get:

(17)

where is the velocity quantum
number, and nr is the space quantum number. Using this in
equation (15) we get:

(18)

Using this in equation (14) we get a connection between the quantum
energies and the space quantum numbers. We can thus write:

(19)

As the lowest energy state of the atom, E1, corresponds to a
space quantum number nr1 = 1, we can write equation
(19) as follows:

(20)

where we get the higher energy states by giving nr2
the values 2, 3, 4, etc. It should be denoted that these energies are
negative, which can be seen from equation (13), and which corresponds to
the bound states of the electron.

If an electron goes from f.i. a higher energy state E2 to the
lower state E1, the difference in energy will be emitted as a
photon. As these two energy states correspond to the space quantum
numbers nr1 = 1 and nr2 = 2,
we can write:

(21)

where Ef gives the energy of the emitted photon. Equation (21)
thus only expresses the demand for conservation of energy.
The numerical value of E1 is equal to the ionization energy, i.e.
the positive energy necessary to transfer to the atom in order to break away
an electron from the energy state E1. In order to calculate the
energy states of the atom it is sufficient to measure the ionization
energy.

We can now write equation (21) in a general way, corresponding to the
space quantum numbers nr1 and nr2.
We get

(22)

In equation (22), Eion is the measured ionization energy for
a real atom. The energy of an emitted photon can now be calculated by
giving the natural numbers m1 and m2 the values:
m1 = 1, 2, etc. and m2 = m1 + 1.

Equation (22) is formally identical to the formulas found by Balmer and
Rydberg on a purely empiric base. They were not able to give a reason
based on the fundamental physical condition of our Universe.

It should be noted that the previous deductions are based on 'point
charges', and this result in not realistic geometrical conditions. In a
more realistic model, it is necessary to take into account the extension
of the particles, which will give another mathematical expression of the
Coulomb energy. Likewise the magnetic energy conditions due to the
electron's and the proton's motions in orbit and their own motions (spin)
must be considered. If these energy amounts are inserted in the calculations,
it will be possible to describe the hydrogen atom's so-called fine structure
and hyper fine structure.

Finally I shall again underline that the previous calculations show,
that the quantum aspects in our Universe are caused by the fundamental
quantization of space, time and mass, and that this quantization is
determined by the total and definite mass of our Universe.